Newspace parameters
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.8077322380\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 9276x^{4} + 17959899x^{2} - 616730624 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 9276x^{4} + 17959899x^{2} - 616730624 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 34300\nu^{3} - 148003099\nu ) / 21420000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 34300\nu^{3} + 469303099\nu ) / 10710000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} + 32200\nu^{2} - 32729896 ) / 2625 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 68\nu^{4} - 284900\nu^{2} - 255241768 ) / 2625 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17011\nu^{5} - 155077300\nu^{3} + 287387477089\nu ) / 4284000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 17\beta_{3} + 309200 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} + 52061\beta_{2} + 359287\beta_1 ) / 300 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 322\beta_{4} + 2849\beta_{3} + 66832504 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 10290\beta_{5} + 30566131\beta_{2} + 293748212\beta_1 ) / 30 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−152.321 | −2014.00 | 15009.6 | 0 | 306775. | 69973.1 | −1.03846e6 | 2.46189e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −127.310 | 2045.53 | 8015.83 | 0 | −260416. | 258383. | 22427.7 | 2.58986e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −40.5284 | −298.988 | −6549.45 | 0 | 12117.5 | −377278. | 597447. | −1.50493e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 40.5284 | 298.988 | −6549.45 | 0 | 12117.5 | 377278. | −597447. | −1.50493e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 127.310 | −2045.53 | 8015.83 | 0 | −260416. | −258383. | −22427.7 | 2.58986e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 152.321 | 2014.00 | 15009.6 | 0 | 306775. | −69973.1 | 1.03846e6 | 2.46189e6 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.14.a.e | 6 | |
| 5.b | even | 2 | 1 | inner | 25.14.a.e | 6 | |
| 5.c | odd | 4 | 2 | 5.14.b.a | ✓ | 6 | |
| 15.e | even | 4 | 2 | 45.14.b.b | 6 | ||
| 20.e | even | 4 | 2 | 80.14.c.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.14.b.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 25.14.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.14.a.e | 6 | 5.b | even | 2 | 1 | inner | |
| 45.14.b.b | 6 | 15.e | even | 4 | 2 | ||
| 80.14.c.c | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 41052T_{2}^{4} + 440779968T_{2}^{2} - 617678127104 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 617678127104 \)
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| $3$ |
\( T^{6} + \cdots - 15\!\cdots\!36 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 46\!\cdots\!44 \)
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| $11$ |
\( (T^{3} + \cdots - 39\!\cdots\!68)^{2} \)
|
| $13$ |
\( T^{6} + \cdots - 52\!\cdots\!16 \)
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| $17$ |
\( T^{6} + \cdots - 19\!\cdots\!24 \)
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| $19$ |
\( (T^{3} + \cdots + 45\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 38\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots + 48\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 18\!\cdots\!12)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 17\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots - 19\!\cdots\!48)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 85\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots - 55\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots - 41\!\cdots\!36 \)
|
| $59$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 18\!\cdots\!68)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 19\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots + 50\!\cdots\!72)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 80\!\cdots\!96 \)
|
| $79$ |
\( (T^{3} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 63\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + \cdots + 20\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 15\!\cdots\!64 \)
|
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